CRYSTAL GROWTH & DESIGN 2005 VOL. 5, NO. 1 3-16
Perspective Toward a Molecular Understanding of Crystal Agglomeration Michael Brunsteiner,† Alan G. Jones,*,‡ Federica Pratola,‡ Sarah L. Price,† and Stefaan J. R. Simons‡ Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ United Kingdom, and Department of Chemical Engineering, University College London, Torrington Place, London, WC1E 7JE United Kingdom Received May 19, 2004
ABSTRACT: A predictive model of the effect of crystal agglomeration on particle form and size distribution requires the quantification of various process parameters that depend on the microscopic properties of specific crystal faces and their interaction with the solvent. In this article, we discuss the various stages in the agglomeration process, using the results of recent experiments on breaking the agglomerative bond and atomic level simulations on the forces involved in crystal aggregation, to highlight the questions that need to be resolved for agglomeration processes to be understood. 1. Introduction During industrial crystallization, nucleation and crystal growth are the primary particle formation processes, but within agitated suspensions, secondary processes, including particle breakage, abnormal growth, and agglomeration, can have a determining effect on product quality,1 in particular, on product form and shape and particle size distribution (PSD). Bimodal PSDs have been observed from well-mixed batch crystallizers2,3 and from continuous mixed-suspension, mixed-productremoval (MSMPR) crystallizers.4-6 Such distributions may be caused either by a size-dependent or dispersed aggregation rate, or by agglomerate disruption, or a combination of the two, but the mechanics of formation are not yet fully understood. Here, we take agglomeration to mean the intergrowth of aggregates formed by particle collisions, through a cementation process that forms an agglomerative bond. Some of the factors involved in particle agglomeration have been demonstrated in an elegant study of the precipitation and subsequent agglomeration of calcium oxalate dihydrate (CaOx) crystals in an MSMPR mininucleator/Couette aggregator sequence.7,8 Agglomeration increased strongly with increasing CaOx supersaturation, while disruption rates decreased, indicating that a change in ionic conditions at the crystal surface * To whom correspondence should be addressed. E-mail: a.jones@ ucl.ac.uk. † Department of Chemistry, University College London. ‡ Department of Chemical Engineering, University College London.
enhanced the probability and strength of particle attachment on collision. Agglomeration was only weakly dependent on agitation rate, while rupture of the agglomerates increased presumably due to increasing turbulence. Both crystal aggregation and agglomerate disruption have been inferred empirically to depend on crystal growth rates and, hence, solution supersaturation. The inference is that the strength of the crystal agglomerates appears to be a function of supersaturation. Thus, crystals appear to both aggregate and form agglomerative bonds at a faster rate at high supersaturation, with the resultant agglomerates becoming more difficult to disrupt.9,10 A critical factor in understanding this behavior is the agglomerate bond strength, both during formation and subsequently, in comparison with the imposed forces due to turbulent motion, which contribute to agglomeration and particle breakage. We report here the results of studies to make a fundamental analysis of crystal agglomeration and to assess what is required for a predictive understanding. First, direct measurements are made of the force of attachment of individual crystals within an agglomerate in a supersaturated solution, as a function of supersaturation and the type of crystal faces, using a microforce balance. These results are then compared with bulk data from continuous crystallizers at various power inputs,11 showing that the agglomerate bond strengths are compatible with the fragmentation of agglomerates. Finally, an understanding of the dependence of the adhesion forces on the
10.1021/cg049837m CCC: $30.25 © 2005 American Chemical Society Published on Web 09/15/2004
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with γ˘ as the shear rate and Lu and Lv as the sizes of the aggregating particles. As the aggregation rate increases with the volume of the particle, it is more likely that a large particle is involved in an aggregation event than a small particle. In similar form, aggregation due to diffusion of particles in eddies in a turbulent flow regime can be expressed as Figure 1. The stages of crystal agglomeration (A) approach of two crystals on a macroscopic length scale, (B) aggregation, where for separations below several nanometers, the microscopic structure of the solvent, solute, and impurities affect the interaction between the crystals, and (C) a solid agglomerative bond forms between the particles.
nature of the crystal surfaces and interface between aggregating crystals is gained through atomic level simulations. This comprises modeling the surface structures, crystal morphology, and the energetics of the interactions between each specific surface and the solute and solvent molecules and between other surfaces. In this way, the propensity for crystals to aggregate and then agglomerate, depending on the nature and orientation of the contact surfaces, can be considered. 2. Crystal Agglomeration: From Crystallizers to Atomic Level Agglomeration occurs when the motion of two or more crystals in the suspension allows them to aggregate (Step 1, Figure 1a,b) and stay together sufficiently long for the growth and intergrowth of an agglomerative bond (Step 2, Figure 1c). After aggregation, there is solution between the crystals, which determines the forces holding the crystallites together while an agglomerative bond of the crystallizing material forms. The liquid between the aggregated crystals is very different from the bulk solution, and indeed, since it has been shown that liquids confined to molecularly thin layers may undergo a liquid-to-solid transition,12 may be able to sustain a shear stress for macroscopic times. By simulating the forces between crystallites separated by a few molecular layers, and examining the nature of the agglomerative bond after breakage, we can make some inferences about the transition from an aggregate to an agglomerate. Further collisions can lead to the breakage of both aggregates and agglomerates (Step 3) (the former may be considered aggregation inefficiency). Various expressions have been proposed to model the rates of both the crystal aggregation and disruption of the agglomerative bond, based on either sound theoretical analysis of the particle motion or purely empirical considerations. These models are described in this section, with particular emphasis on the assumptions and empirical parameters required to represent the forces between the crystallites prior to and after the formation of the agglomerative bond. 2.1 Step 1: Aggregation and Collection Efficiency. As early as 1917, Smoluchowski13 showed that the rate of aggregation of spherical particles in a laminar shear field can be expressed as
1 Kagr(Lu,Lv) ) γ˘ [Lu + Lv]3 ) βagr[Lu + Lv]3 6
(1)
Kagr(Lu,Lv) ) kU[Lu + Lv]3 ) βarl[Lu + Lv]3
(2)
with U as the velocity gradient of the fluid in turbulent motion being proportional to 1/2, where is the energy dissipation rate14 (see later). In this model, the crystals are assumed to be spherical and to aggregate on contact. Thus, the applicability of this model will depend on the actual morphology and whether the forces between the crystals for all possible contacts are strongly attractive. If we consider this process at the molecular level, it describes the approach of two particles on a macroscopic length scale (Figure 1a). However, once the particles are within several nanometers (Figure 1b), the forces between them are determined by the specific nature of the surfaces of the particles and the nature of the solution phase between them, as well as the dynamics of the approach. A mean field hydrodynamic model15 of aggregation to account for aggregation efficiency predicts a size dependence of aggregation that passes through a maximum with increasing mean shear rate, together with a correlating parameter for the aggregation efficiency based on the ratio of aggregate strength to applied force between particles. Mersmann and Braun16 and Hounslow et al.17 provide detailed reviews of work in this area. Thus, the aggregation rate, β, can be divided into three components18:
β ) Weff βcoll β/L
(3)
where βcoll reflects the collision rate of the particles, β/L is the size dependency and Weff is the proportion of collisions that result in an aggregation, which should be determined by the chemical nature of the surfaces and solvents. In this paper, we use molecular dynamics simulations to demonstrate some of the factors that influence Weff. 2.2 Step 2: Formation of the Agglomerative Bond. Very little has been established about how the crystal faces, once in contact, actually form an agglomerative bond, although liquid bridges play an important role.19,20 Other evidence comes from the decrease in the disruption rate of calcium oxalate9 and of calcium carbonate10 with increasing growth rate. When agglomeration takes place during precipitation, agglomerates are only recently attached, so the crystalline bridges between the primary particles are not yet completely desolvated and cemented and are therefore in a viscous liquid phase. The bonds of some agglomerative fragments can thus be broken relatively easily by agitation. Presumably, primary particles would usually (although not necessarily invariably) be the easiest to remove by either fluid shear or collision. On the other hand, previously aged and dried crystal agglomerates, often used as seeds, are mechanically stronger10 and should therefore be more difficult to break and so their attrition occurs at a lower rate.
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Understanding the growth and formation of the agglomerative bond requires considerable insight into its nature. Some constraints on the atomic structure of the bond arising from the nature of the faces are discussed in section 4.1. The formation of a bond as a growing neck, as illustrated by Figure 1c, is one model15,17 of the cementation process, and some considerations of factors that will affect its strength are considered in section 4.2. However, examination of the surfaces of broken agglomerative bonds, described in section 3.3, show that, at least for the systems studied, the single neck model is inappropriate, and the structure of the agglomerative bond is more complex, with the inclusion of solvent. 2.3 Step 3: Disruption Kernel. The function most often used in modeling breakage and disruption processes has the form
Kdisr(Lu,Lv) ) βdisr[Lu + Lv]3
(4)
With this kernel, the disruption rate is proportional to the particle volume. This theoretical assumption was validated by Synowiec et al.,21 who demonstrated that a third-order dependence on the particle size (and, therefore, proportionality on particle volume) can be explained by the dominant disruption mechanism of turbulent crystal-shear forces. The disruption rate is also a function of the degree of supersaturation prevailing in the reactor. High supersaturation results in high growth rates and strong agglomerates, and thus reduces breakage.9,10 In addition, the rate of disruption increases with increased power input. Therefore, the disruption rate function becomes
Kdisr(Lu,Lv,,S) ) βdisrrSsf (Lu,Lv)
(5)
where S is the supersaturation ratio, defined later by eq 10. These expressions are dependent on the strength of the agglomerative bond. Experiments on the direct measurement of this bond strength, required for the βdisr model, are described later in section 3. Thus, we can predict how agglomerated crystals are broken down within crystallizers, and a comparison of the fragment size after disruption is discussed in section 3.3. 2.4 Overall Agglomeration Rates. This complex process of aggregation collisions (with the forces that operate over the last few nanometers of the collision determining the conditions for the formation of the agglomerative bond) and then its potential breakage, gives a net agglomeration rate of bond making and breaking. An empirical formula of a kernel22 that includes the particle size factors is
[Lu3 - Lv3]2 (u - v)2 Kaggl(Lu,Lv) ) CAEi ≈ βaggl u+v L 3+L3 u
(6)
v
where u and v denote the volumes of the particles, CA is a collision rate parameter, and the collection efficiency parameter, Ei, absorbs the system dependent factors discussed above. Hartel et al.7 and Hartel and Randolph8 successfully modeled the agglomeration kinetics of calcium oxalate using this kernel. Thus, CA is
comparable to Weff (eq 3) if disruption is negligible and both are empirical factors for the effectiveness of collisions in producing agglomerates. Despite theoretical considerations (eqs 1, 2, and 6) implying a positive size-dependence of agglomeration, the PSDs observed can often be approximated by a sizeindependent kernel
Kaggl ) βaggl * f (Lu,Lv)
(7)
This apparent size-independence of the agglomeration rate may occur because both aggregation inefficiency23 and particle disruption8 increase with particle size. These dispersive processes may counteract the positive effect of aggregation, thereby imposing agglomerate particle size limitations. Thus, in a mixed system, as, for example, in a stirred tank, the rate of agglomeration additionally depends on the shear field and, therefore, on the energy dissipation in the tank. Camp and Stein24 originally proposed that the mean fluid shear rate (velocity gradient) during particle flocculation is proportional to the square root of the mean energy dissipation
h˘ ) γ
xµj
(8)
In reality, the energy dissipation distribution within the vessel will lead to a marked variation in the particle collision velocities throughout the vessel.25 In crystallization systems, solution supersaturation also plays an important role, as the higher the supersaturation, the “stickier” the particles and the easier they agglomerate.26 This leads to a general empirical formulation of the agglomeration rate27
Kaggl(Lu,Lv,,S) ) βagglpSqf (Lu,Lv)
(9)
where both the energy dissipation and the level of supersaturation are accounted for using a power law function and an efficiency of agglomerative bond formation and disruption following collisions is encapsulated in βaggl. It can be seen that we have a variety of macroscopic expressions and observations on agglomeration that approximate or parametrize the nature of the collisions, the forces between the interacting crystals and the force needed to disrupt the agglomerative bond. The latter we can study experimentally, whereas insights into the former can only be gained through computer simulation. These experiments and simulations are described in the next two sections. The results give some insights into the factors that need to be considered to develop a molecular understanding of agglomeration and a method of predicting agglomeration rates for a specific system. 3. Experimentation al.11,28
Pratola et describe a novel microscopic method for determining the strength of agglomerating crystals in supersaturated solution (Figure 2) based on the microforce balance (MFB) technique of Simons and Fairbrother.29 The experimental equipment consists of an Olympus IX50 inverted optical microscope, fitted with an adapted stage, on which two micromanipulators
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Figure 2. Schematic view of the MFB stage and micromanipulators: A stage, B and C micromanipulators, D objective, E temperature controller, F flexible blade, G travelling platform, H travel piezo adjuster, I and L LVDT.
Figure 3. Potash alum (111) crystal surface (a) a flat surface showing fringes of interference and (b) a rough surface not showing fringes of interference.
are employed to hold a pair of micropipets. This novel apparatus can measure adhesion forces between microscopic particles down to 10 µm in diameter. The microscope and a video camera are positioned under the stage, leaving the space above clear for operation of the manipulators. The special design of manipulator B (see Figure 2) allows free rotation of the supported pipets, which can be oriented perpendicular to the stage, to mount the crystals onto their tips, and then moved onto the manipulators for measurements to be carried out. With this tool, it is possible to have two crystals with opposing parallel faces. Before measurements, the pipets supporting the two selected crystals are fixed onto the manipulator parallel to the stage with their tips converging. The supported crystals are then brought into contact. By using a travelling platform on the stage G, an optical dish containing a presaturated solution can be raised to submerge the crystals. The agglomeration occurs under a defined supersaturation level attained by controlling the temperature with the annular thermostatic bath E. The strength of the agglomerate can be assimilated to the force needed to disrupt the agglomerate. During measurements, a long travel piezo adjuster H (MDE 227, Elliot Scientific) imparts a controlled movement to the manipulator C. The force necessary to break the agglomerate is measured by using two linear variable differential transducers (LVDT): LVDT I (type GTX 1000) detects the movement of the manipulator C, while LVDT L (Type G5, RDP electronics) detects the flexion of a calibrated flexible blade F (a dynamometer suitable for solid bridges), mounted on the manipulator B. The LVDT L allows the measurement of the flexion of the blade (and hence the agglomeration force). A video camera (JVC 55) connected to the microscope captures and records images of the crystals before and after rupture. Crystal contact area and geometry are measured using an image analysis technique. All electronic devices on the MFB are computer controlled, with the force and displacement measurements saved to hard disk. 3.1 The Crystal-Solvent Systems. Studies were performed using potassium aluminum sulfate (potash alum) and succinic acid in aqueous solutions. The crystallization of potash alum (K2SO4‚Al2(SO4)3‚24H2O, M ) 948.8, F ) 1760 kg m-3) from aqueous solution has been extensively studied by a large number of workers over the past 20 years.30-32 This substance is often selected in investigations on crystallization because (i)
it invariably crystallizes in the form of discrete, nearperfect octahedra (with a surface shape factor of 2.46), (ii) it has well-known kinetics of crystallization, and (iii) it has high solubility that gives a wide metastable range avoiding excessive and unwanted nucleation. Furthermore, of advantage to this study is the fact that potash alum exhibits three distinctive types of crystal faces (111), (110), and (100), which enables the effects of the nature of the opposing faces at the bond site on the agglomerate bond strength to be investigated. A smaller set of experiments was performed on an organic crystal, succinic acid, to investigate the mean value of agglomerate strength under a fixed supersaturation condition and the effect of supersaturation on agglomerate strength. Experiments were performed using (020) faces only, since the experimental morphology of succinic acid crystals are mainly “monodimensional”. The crystal surface topography was assessed by means of optical interferometry to detect imperfections on the free surfaces of each crystal.33 The technique can give information about the surface microtopography of a crystal by the application of localized multiple-beam fringes to the surfaces.34 To reveal fringes of interference on a crystal surface for the examination of the surface microtopography, the inverted Olympus IX-70 was used as an interference microscope focused on crystals sprinkled onto a glass slide. The appearance of interference fringes indicates a smooth and flat surface. Hence, those crystals exhibiting such fringes have smooth faces lying parallel to the glass microslides and can be glued onto the micropipets following the procedure described above, to ensure measurement of agglomeration forces between crystals with parallel faces. In Figure 3a the microinterference pattern is shown under ×10 magnification on the (111) surface of a potash alum crystal. The surface of the crystal is seen in focus and is covered with interference fringes. The distinct discontinuity shown in the pattern of the fringes is due to a cleavage step. The comparison between Figure 3, panels a and b underlines the different aspects shown by two crystal surfaces with and without the appearance of interference fringes, respectively. The surface image quality is enhanced by the contrast in brightness of the fringes and enables the identification of the surface type. In the specific example of Figure 3a the characteristic shape of a (111) surface is clearly noticeable. The interferometric technique, successfully used to assess the flatness of the potash alum surfaces, was
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Crystal Growth & Design, Vol. 5, No. 1, 2005 7 Table 2. Experimental Conditions of Experiments Performed to Measure the Agglomerative Strength of Different Pairs of Succinic Acid Crystal Faces in Contact conditions contact time saturation temperature dimensionless supersaturation spring constant
45 min 25 °C 0.16-0.35 186 N m-1
Figure 5 as a function of the dimensionless supersaturation level, calculated as: Figure 4. Succinic acid surface not showing interferometric fringes. Table 1. Experimental Conditions of Experiments Performed (a) To Assess the Effect of Supersaturation on Agglomerative Strength for Potash Alum (b) To Measure the Agglomerative Strength of Different Pairs of Potash Alum Crystal Faces in Contact (a) Experimental Conditions saturation temperature 25 °C contact time 45 min dimensionless supersaturation 0.08-0.19 faces in contact (111) only spring constant 342.77 N m-1 (b) Experimental Conditions saturation temperature 25 °C contact time 15-20 min dimensionless supersaturation 0.12 faces in contact (111), (110), (100) spring constant 342.77 N m-1
found to be ineffective for succinic acid. As emphasized in Figure 4, which depicts a succinic acid (020) face taken under the microscope using the interferometric technique, only a few fringes of interference are apparent. This suggests that succinic acid crystal faces are not sufficiently flat and regular to form fringes of interference over the extent of the surface. As a consequence, the choice of the surface to be used for experiments was performed by visual observation under the microscope. 3.2 Specific Experimental Procedures and Results. 3.2.1 Effect of Supersaturation. Experiments were performed to investigate the effects of supersaturation on agglomerate strength using a solution of potash alum saturated at 25 °C.35 The solution was stored at around 30-35 °C and kept agitated using a magnetic stirrer. The experimental conditions are summarized in Table 1a. The experiments were performed using (111) crystal faces of potash alum only. For each experiment, a pair of regular and flat crystals was selected under the microscope using the interferometric method, glued onto the two micropipets, and then fixed on the manipulator with their tips converging, as described in the previous section. A total of 3 mL of the prepared solution was poured into the thermostatic dish. The two crystals were brought into contact and then immersed in the solution and left for 45 min to agglomerate. After this period of time, the agglomerate was broken and the displacement of the two LVDTs were recorded. An analogous set of experiments was performed on the (020) crystal faces of succinic acid, with the experimental conditions summarized in Table 2. The agglomerate adhesion force per unit of contact area measured in this set of experiments is plotted in
S)
c - cs cs
(10)
where S is the supersaturation, c is the supersaturated solution concentration, and cs the equilibrium saturation concentration at the prevailing temperature. The results of these experiments are highly reproducible, providing relatively small estimated errors. For example, the mean value of agglomeration strength of succinic acid crystals measured at constant supersaturation (S ) 0.16) is 7.23 × 103 N m-2, with a standard deviation of 2.51 × 102 N m-2. As can be seen in Figure 5, the force per unit area increases with increasing supersaturation level, both for potash alum and succinic acid. It can be observed that, for the same supersaturation level, agglomerate strength for the organic crystal is 1 order of magnitude lower than for the inorganic crystal. In addition, the effect of supersaturation is greater in the latter case. The results presented in Figure 5 are consistent with the evidence shown by different authors that crystal agglomerates are more difficult to disrupt at high supersaturation. To explain the observed dependence of the agglomeration and disruption kernel function on supersaturation, it is often proposed that agglomeration in supersaturated solutions is a two-step process: first crystals must collide; they then must be cemented together.36-38 At high supersaturation, particles become more “sticky”26 so that when they collide they can cement more firmly together and become more difficult to disrupt. Our simulations (section 4.3) show that the supersaturation will have a profound influence on the dynamics of the last few nanometers of approach, as well as affecting the growth rate and microstructure of the agglomerative bond. 3.2.2 Breakage and Rupture of the Agglomerative Bond. During the experiments, sequences of agglomeration and rupture were captured to follow the
Figure 5. Succinic acid and potash alum agglomerate adhesion forces per unit of contact area as a function of dimensionless supersaturation.
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Figure 6. The agglomerate evolution (a) soon after pouring the solution, (b) after 45 min, and (c) agglomerate rupture sequence for potash alum crystals.
Figure 7. Potash alum (111) crystal surface (a) before agglomeration and (b) after rupture, and (020) succinic acid surface (c) before agglomeration and (d) after rupture.
evolution of the agglomerate’s growth (Figure 6a,b) and subsequent rupture (Figure 6c). Images of the crystal surfaces that came into contact were captured before and after each experiment (Figure 7). From the comparison of the two pictures, it is possible to elucidate the area of contact between the two crystals. At the beginning (Figure 7a) the entire surface of the crystal appears to be flat, as is apparent from the interference fringes. However, after the rupture of the agglomerate (Figure 7b), only the surrounding part of the surface is flat, while the central part comprises a rough rim of broken agglomerative bond surrounding a smoother concave surface. The area enclosed by the rough rim can be considered to be the contact area and can be measured using image analysis. These can be contrasted with the images of the surface of succinic acid crystals before agglomeration (Figure 7c) and after rupture (Figure 7d), where the agglomerative bond seems to have formed in patches between the two crystals. The contact area comprises an irregular pattern of smooth depressed and roughened raised areas, the latter being the broken agglomerative bond. 3.2.3 Effect of Different Crystal Faces. In the second set of experiments (Table 1b), the agglomeration behavior between different crystal faces was investigated and the agglomerate strength reported as a function of relative growth rate, calculated for each
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Figure 8. Agglomeration strength as function of faces growth rates.
crystal face as the ratio between the actual and the (111) face growth rate. The crystals were prepared in the manner discussed above, with the type of crystal faces to be brought together being determined by image analysis. Crystals were mounted on the tips of the pipets and only pairs of the same type of faces were used to form the agglomerate bonds. Solution saturation and crystal contact time were kept constant for each face pair. Figure 8 shows that the agglomerate strength per unit face area agglomerative bonds was higher for the smaller faces than the large face. Since the large face can be assumed to have the slower growth rate, this result is in line with the experimental finding that repulsive and attractive forces developed between crystal faces in contact via liquid bridges39 are respectively smaller and larger for faster growing faces than for slower growing faces. However, it should be noted that Figure 7b shows that the actual rough area of broken agglomerative bond has a much smaller area than the crystal face. 3.3 Interpretation of the Experimental Results and Application to Agglomerate Breakage in Crystallizers. Data on crystal breakage collected in MSMPR crystallizers can be used to validate the reliability of the results presented in Figure 5. The agglomerate adhesion force can be related to the size of fragments that are broken away from crystals in an agitated suspension. The breakage of crystals is determined by two opposing factors, namely, the mechanical strength of crystals and the applied breaking forces. The mechanical strength of a crystal can be assimilated to the agglomerate adhesion force measured in our experiments, and depends both on the crystal-solvent system and supersaturation. The breaking force acting on crystals depends on the physical and thermodynamic properties of the agitated solution. The total rate of fragments generated in a stirred vessel can be expressed as the sum of the rate of generation by means of impact and the rate of generation by means of turbulent fluid forces. To determine the energetic contribution of a particular mechanism of crystal break-up to the generation rate terms, the impact (Fim), pressure (Fpr), drag (Fdr), and shear force (Fsh) have to be calculated, as suggested by Synowiec et al.21 The maximum size of fragments (Lfr max) in an agitated suspension of parent (agglomerated) crystals of size L0 can be obtained from the ratio of the calculated breaking force acting on the crystals (F) and the
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Figure 9. Fragment size as a function of parent crystal size for different agglomerate adhesion forces from calculation (lines) and experiment (shaded areas).
measured agglomerate adhesion force per unit contact area (φa):
Lfr max )
() F φa
1/2
(11)
where the maximum fragment size is assumed to be related to the contact area between the fragment and the parent agglomerate as:
A ) Lfr2
(12)
Values of fragment size calculated through eq 11 for different supersaturation levels (correlated to the agglomerate adhesion forces using the experimental data plotted in Figure 5, which shows the force per unit contact area of the crystal “parent” agglomerates) are plotted in Figure 9 as a function of the parent crystal size L0. The estimated fragment size decreases with increasing supersaturation, since, as shown in Figure 5, the agglomerate mechanical strength increases with increasing supersaturation. Estimated values are comparable to those determined experimentally by different authors, as represented in Figure 9 by the shadowed areas. Synowiec et al.21 report that the breakage of potash alum crystals of average size ranging from 107 to 925 µm, suspended in an agitated vessel, resulted in the formation of fragments of size ranging from 3 to 15 µm. Slightly larger fragment sizes (15-17 µm) were measured by Fasoli and Conti40 in an agitated solution of CuSO4‚5H2O crystals of welldefined size (in the range 355-1347 µm) suspended in a nonsolvent solution. Crawley et al.,41 in a study of potassium sulfate crystal disruption in a stirred vessel, found fragments of 5 µm were produced by breakage of parent crystals of 650 µm diameter in an ethanol solution. Several factors may affect the estimation of fragment size through eq 11, because of uncertainties in the calculation of mechanical stresses on particles in agitated systems (F) and in the measurement of agglomerate strength (φa). In particular, two effects should be considered to account for the discrepancy between fragment size estimates and experimental data observed in Figure 9. First, according to the agglomeration mechanism discussed above (i.e., particle collisions followed by
cementation), only if colliding particles remain in contact for long enough will sufficient cementation occur to form a stable agglomerate.36 As a consequence of the time-dependent nature of the bond formation, any agglomerates in intermediate stages of formation will be easier to disrupt. In the experimental setup used here, crystal contact time (45 min) is far longer than that expected in any agitated suspension, and this could lead to underestimates of the fragment size compared to those found experimentally (due to the higher adhesion forces developed during this period). On the other hand, the contact time required to achieve a stable agglomerate will depend on several factors, among which the transfer rate of solute molecules from the bulk solution to the crystals could play a limiting role. Liquid-solid mass transfer rate depends on both supersaturation (related to the transport driving force) and hydrodynamic conditions around the crystals (influencing the mass transfer coefficient). In the experimental setup used here the transport of solutes from the bulk of the solution to the crystals is by diffusion, while in an agitated system this will be by turbulent eddies. Second, according to eq 2, the discrepancy between estimates shown in Figure 9 and the experimental data may be due to an overestimation of the mechanical stresses on the agglomerates, as not all their components will be active for agglomerate breakage. For a given agglomerate strength (φa), a smaller breaking force (F) corresponds to smaller estimates of fragment size (Lfr max) through eq 11. Thus, the comparison made in Figure 9 is only expected to show agreement for the orders of magnitude of the measured forces. 4. Atomistic Modeling of Surfaces and Their Interactions 4.1 Identifying the Crystal Faces Involved in Agglomeration. While the faces of a crystal can be identified from the crystal morphology, establishing the precise atomic structure of the surface, for all but the simplest materials, is more problematic. The simplest model for crystal morphology, the BFDH model of Bravais, Freidel, Donnay and Harker,42 assumes that the slowest growing, and hence largest faces, are those with the greatest interplanar spacings dhkl. This model can be used for identifying which faces should be considered in morphological and surface studies, but is only successful for predicting the morphologies when there is a good correlation between the interplanar spacings and the strength of the interatomic interactions. This is generally poor for organic crystals, where there is considerable difference in the strength of the covalent bonds and the different, highly directional, intermolecular forces that form the crystal. This is exemplified for succinic acid in Figure 10, where it can be seen that the dominant (020) face used in the experiments is a relatively minor face by the BDFH model. The effect of the differing strengths of the atomic interactions can be taken into account in the attachment energy model for crystal morphology. In this model,43,44 the relative growth rates of the faces are assumed to be proportional to their attachment energies, Eatt, defined as the energy released when one additional growth slice of thickness dhkl is attached to the crystal
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Figure 11. The molecular structure of the (020) face of succinic acid crystals grown from the vapor, as predicted by the attachment energy model.
Figure 10. Experimental (vapor grown) and calculated morphologies of β-succinic acid, using the BFDH model,42 which only considers interplanar spacings, and the attachment energy model, which considers the strength of the intermolecular interactions with various models for these interactions.45
face identified by the Miller indices hkl. Attachment energy model predictions for succinic acid (Figure 10) are fairly typical45 of this widely used model for the morphologies of organic molecules. The dominant (020) face is predicted using quite a variety of different models for the intermolecular forces.45 However, all the more accurate models for the intermolecular forces predict that the (111) faces are too large and the (011) faces are not present. This reflects one deficiency in the attachment energy model, that it is only appropriate for the growth rate from vapor of F faces, (i.e., faces that contain two different chains of strong intermolecular interactions), below their roughening temperature, when they grow by a layer by layer birth and spread mechanism.46 The (111) face of succinic acid is not an F face and, hence, the model incorrectly estimates its growth rate and morphological importance. The attachment energy predictions (Figure 10) show sufficient similarity to the experimental vapor grown morphology44 that it is worth examining its prediction of the atomic structure of the (020) face used in the experiments (Figure 11). The carboxylic acid groups dimerize through hydrogen bonds, with the different chains of molecules arranged so that there are close interactions between the carboxylic acids dimers, thus confirming that it is an F face. The carboxylic acid groups are essentially in the plane of the surface, and it is the less polar CH2 groups that protrude, further limiting the possibility of other molecules being able to attach themselves to the surface by hydrogen bonds. The crystals obtained from solution and the experimental
examination of this face show that it is rough on the scale of the wavelength of light. This suggests that the surface in Figure 11 is a poor representation and that there is surface roughening. The nature of the surface in Figure 11 does suggest that water molecules may well affect this surface by interacting with the carboxylic acid groups. The attachment energy model can fortuitously give good results for high symmetry crystals such as potash alum, as it only requires relative growth rates of the few observed faces. In this case, the octahedral geometry is reproduced, along with the (100) faces, but the calculations do not predict the minor (110) faces seen on some crystals,47 as in Figure 9. The method does not allow for surface relaxation, and predicts that pairs of polar surfaces grow at the same rate (whereas a differential growth rate is apparent in the urea morphology48) and does not allow for the effect of solvent. Hence, it is totally inappropriate for determining the likely structures of the faces of potash alum crystals within solution prior to agglomeration. To get an atomistic model of potash alum, we performed molecular dynamics simulations of various faces (100), (110), and (111) in contact with saturated aqueous solutions. The key problem was for the dominant (111) face, as there are four simple ways in which this face can be cut, exposing different sets of ions (Figure 12), but all of them result in a net dipole perpendicular to this face. Such polar surfaces are a major topic in surface science research because of their intrinsic instability. Several mechanisms can operate to eliminate, or at least reduce the surface dipole. There can be rearrangement from the perfect cut crystal surface configuration, ranging from relaxation of the surface ions to gross reconstruction (for example, half the ions in the surface layer effectively relocated on the reverse surface of the crystal), to mesoscale rearrangements providing microfaceting. Alternative mechanisms are the attachment of counterions and other polar molecules by physisorption, through to chemisorption of solvent
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Figure 12. Atomic structure of potash alum surfaces, showing plausible cuts from the perfect crystal. Each slab contains four unit cells and is viewed from the side, with the surfaces on top.
ions and impurities, or modification of electronic structure of the surface. For potash alum, electronic structure modification can be ruled out, although chemisorption of hydroxyl groups or protons from water dissociation cannot be eliminated. Photographs49 of a screw dislocation on the (111) face of potash alum suggest that this surface is flat for approximately 0.2 µm, so mesoscale faceting is unlikely. The other possibilities, physisorption, surface relaxation, and four possible reconstructions were investigated by molecular dynamics50 as part of this project. The complexity of the potash alum (K2SO4‚Al2(SO4)3‚ 24H2O) crystal and its interface with a saturated solution makes it very challenging to represent the forces between the atoms adequately. Although atomistic modeling of simple ionic crystals and their solutions is well developed,51 this study50 is among the earliest into hydrated ionic crystals. Hence, by adopting a commonly used rigid water-water model52 and an existing flexible sulfate-sulfate model,53 together with empirical adjustments of the potassium (K+) and aluminum (Al3+) ion interactions, we eventually obtain an isotropic atom-atom point charge model, representing the repulsion, dispersion, and electrostatic interactions. This model potential reproduced the crystal structure at 300 K satisfactorily (cell length within 0.2%), including the disorder of the sulfate groups, estimating 5% disorder in contrast to experimental estimates of 10-30%.54 The molecular dynamics calculations were carried out at a nominal temperature of 300 K using the program DL_POLY55 on structures of 2 × 2 surface unit cells, with a depth of 3-4 unit cells in the crystal slab. Only three sublayers of ions were given the flexibility to relax under the influence of a solution of 576 water molecules, including six SO42-, three Al3+, and three K+ ions to represent the saturated solution. There was a vacuum (gas) phase section above the solution. The possible positions for cutting the crystal to approximate each
Figure 13. Structure50 of the (100) and (110) surfaces, and one plausible model of the (111) surface of potash alum in saturated solution (as shown by a snapshot of the simulation) and time averaged distribution of the ions in the direction perpendicular to the surface. The positions along the surface are given in Å, with a brace indicating corresponding positions in the two diagrams. The crystal atoms below the thin dashed line are held rigid and the thick dashed line denotes the surface. The water molecules within the crystal structure are omitted for clarity, as very few diffused into solution. The peaks in the density distribution plots have been normalized so that their maximum values are approximately equal, for ease of comparison.
surface shown in Figure 12 were investigated. For the polar (111) surfaces, four reconstructions to give nonpolar neutral slabs were investigated, as well as dipolar and charged slabs with various methods of neutralizing the net dipole or charge.50 Although the system size and length of simulation time (300-600 ps) are very small as seen from the perspective of macroscopic agglomeration studies, such computations have only recently become feasible due to advances in computer power. The results for the (100) and (110) surfaces show that the proposed surfaces are stable (Figure 13). The snapshot shows that there is some disorder in the surface layers, with one K+ ion dissolving into the solution above the (100) surface. The time-averaged density distribution functions show that the amplitude of motion of the atoms in the surface layers are much as would have been expected from a stable surface at room temperature, with even the uppermost sulfate ions in the (100) surface only moving up to 2 Å from their lattice site. The solution layer shows a tendency for the K+ ions to get closer to the surface than the Al3+ ions,
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because the hydration shell around the Al3+ is very persistent. The simulations are consistent with the experimental observation56,57 that the exchange rate in the K+ hydration shell is about eight orders of magnitude faster than in the Al3+ shell. The simulations of both the potassium layer terminated surfaces (K-t and KS-t in Figure 12) showed a clear tendency to dissolve. Reconstructed Al terminated surfaces with half the Al3+ removed to neutralize the dipole, tended to attract K+ ions out of the solution into the empty Al pockets and they persisted in this noncrystallographic position for sufficiently long that it seems likely to inhibit crystal growth. Thus, it seems very probable that the potash alum (111) surface is terminated by an Al SO4 layer, and one of the more plausible variations50 is shown in Figure 13. The plausible surfaces appear stable, with only reasonable amplitude motions relative to the considerably relaxed surface structure. They differ by the amount of sulfate in the surface, which is likely to be determined by the sulfate composition in the solution. Common to all stable models of a (111) surface is a strong layering of the cation layers in the solution, with the K+ moving closer to the surface, but also the Al3+ acting to shield the surface. This layering of the ions above the surface is compatible with easy crystal growth. The diffuse nature of the surface, with ions layered above a somewhat relaxed and disordered crystalline surface, inevitably makes the position and net charge on the surface rather ill-defined quantities once the crystal surfaces approach within about 20 Å. A charged surface does not contradict the high tendency of potash alum to agglomerate, because a number of mechanisms exist that can promote attraction between equally charged surfaces in solution,58 particularly if there are divalent counterions in the solution. 4.2 Nature of the Agglomerative Bond. The most ideal and maximum strength structure for the agglomerative bond is for it to have grown as a perfect continuation of the crystal structure. This is only even hypothetically possible for two identical surfaces, perfectly aligned on the atomic scale. In this case, the energy to sever the agglomerative bond would be twice the surface energy per unit area of the bond. The surface energy per unit area, σ, is defined as
σ ) (E surface - E bulk A A )/A
(13)
is the energy of a region of the surface A where E surface A is the energy for the same number of unit and E bulk A cells in the bulk, and may therefore be derived from the lattice energy. This can be estimated for a surface (in contact with vacuum) by most of the computer programs that are used for estimating the attachment energy, as it is essentially the attachment energy for an infinite thickness slice, provided the face is not polar. When the two surfaces are different, an ideal perfect crystal growth cannot occur. We developed48 the modeling program ORIENT59 to investigate this, by considering the interaction between a small nanocrystallite of one urea surface interacting with an infinite surface of the other urea face in the xy plane. The nanocrystallite of size Nx × Ny and thickness Nz unit cells (where Nz was chosen to ensure convergence) was allowed to
optimize its position over the other infinite surface. In the case of a urea (001) face in contact with a (110) face, the interaction area energy per unit area converged remarkably quickly with n ) Nx ) Ny to -0.202 ( 0.001 J m-2, an energy for this optimized interaction that is only 20% smaller than for the perfect crystal growth of the (110) face and about 50% smaller than that for the smaller (001) face. The two surfaces obviously align the zone directions. There are different minute shifts of the nanocrystallite in the nonconformable direction as the size of the nanocrystallite varies, which result in the apparent convergence of interaction energy per unit area with size of the idealized urea crystallites attached to the urea surface. This suggests that an agglomerative bond could grow between the unlike surfaces that would be only slightly distorted from a perfect crystal structure and thus relatively strong. The above methodology was designed to detect whether two surfaces were so incompatible in structure that any agglomerative bond would necessarily be amorphous and, therefore, considerably weaker. It was based on the neck model for the cementation of highly idealized nonpolar crystal surfaces in a vacuum, and so could not be applied to potash alum. The simple model for estimating the compatibility of unlike surfaces emphasized that the relative orientation of the surfaces as they came into contact, as well as the details of the surface structure, would play a major role in determining the potential nature of the agglomerative bond. Clearly, more information is required as to the nature of the forces between simple atomic crystal surfaces, before we can quantitatively introduce the additional complexity of the required alignment of more complex (organic) surfaces (such as Figure 11). 4.3 The Forces Acting on Agglomerating Particles. The forces acting between the surfaces of two particles in solution will only resemble the smooth curve of the DVLO theory,60 while the separation is so large that it is appropriate to consider the solvent as a continuum. As the particles approach distances more comparable to the extent of the ordering of solvent ions or molecules above the surfaces (as seen in Figure 13), then the interaction and disturbance of these layers will produce significant variations in the forces. As the separation decreases further, the actual size, shape, and interactions of the molecules will become important and the forces may oscillate from being attractive to repulsive. These oscillations, with regions of repulsive forces, can form a barrier to the effective collision and subsequent agglomeration. If we regard the collision and agglomeration as a second-order reaction, then the activation free energy barrier ∆G* can be obtained by integrating the average force between the surface (averaged over solvent effects) from infinite separation up to the contact distance for agglomeration. To study these forces, we have performed molecular dynamics simulations on the system shown in Figure 14, with two idealized nanocrystallites immersed in a solution in contact with a gas phase region. The large solution bath and gas phase is to allow the system to equilibrate to approximate constant chemical potential, by exchange of solution ions in the region between the crystallites and the bulk solution. The only other simulation61 of forces in nanopores that explicitly mod-
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Figure 14. System geometry for the molecular dynamics simulation62 of a slit pore in contact with bulk solution. Lx ) 32.2 Å, Lz ) 129.95 Å, dc ) 18.84 Å, wc ) 22.2 Å, ds ∼ 20 Å. The nanocrystallites are either held in perfect register (i.e., the xy coordinates fixed so that if dp ) 0 the two nanocrystallites would correspond to a perfect crystal) or allowed to adjust their position in response to the forces.
eled ionic solutions used continuum methods to estimate the density of ions and solvent at each separation of the surfaces. As we shall see later, the uncertainty in either of these methods in estimating the equilibrium concentration of solvent between the surfaces separated by less than 1.5 nm may be considerably less than the uncertainty in whether the liquid between two nanocrystallites coming together in a crystallizer is in equilibrium with the bulk at these separations. The system chosen for the simulations62 was the (110) face of potassium chloride, a well-studied simple ionic system. The nanocrystallites had a surface area of 5 × 5 unit cells and the ions in the nanocrystallites were held rigid in the perfect crystal structure, while the solution atoms were equilibrated for over 300 ps, followed by a simulation of about 400 ps, over which the forces acting on the nanocrystallite were averaged both over time and the nanocrystallite surface. The statistical errors in the measured forces, indicated by the errors bars, could have been reduced by using longer simulation times, but this was not appropriate given the considerable computer time required for these initial investigations. The results in Figure 15 show that for two nanocrystallites fixed in perfect register (Figure 14) with their faces separated by approximately 6.6 Å, the total forces from the interactions of the crystallites and the water are repulsive. When this separation is reduced to approximately 6 Å the forces are attractive. At 6 Å, the water molecules can form a structured layer between the two surfaces, as shown in Figure 16a, and so add to the attractive forces between the nanocrystallites. However, at other separations the fit is not so good, giving broader, disordered distributions of water molecules between the surfaces. This produces repulsion, although the forces become attractive again when the water in the pore can order itself into a double layer around 6.8 Å. When KCl is added to represent a saturated solution (117 KCl to 1292 water molecules), the forces are significantly changed. In the solution at 6.0 Å (Figure 16b), the one Cl- ion remaining in the pore after equilibration disrupts this structure locally, diminishing the attraction. If the crystallites are allowed to move laterally in response to the shearing forces from the molecules in the pore, this considerably modifies the forces as it allows the solution and crystallite to adjust to a more
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Figure 15. The total forces on the nanocrystallites of KCl as a function of separation. The crystallites are in the simulation cell shown in Figure 14 and each has a surface area of 492.8 Å2. The forces are shown when the crystallites are held rigid in perfect register and also when the crystallites are allowed to move laterally, at the fixed separation, in response to the shear forces resulting from the liquid structure. Simulations for the crystallites in water and also in a saturated solution are shown. The lines merely join the simulated points. The error bars indicate the statistical errors in the simulation.
favorable structure. When there is just water in the pore at 6 Å this has little effect, but with the KCl solution (Figure 16c) there is a huge difference, as the surfaces move so that positive ions are above each other, separated by negative ions in the solution and vice versa. These results all assume that the approach of the two surfaces is sufficiently slow that the water and solute concentration between the surfaces comes to equilibrium. If we forcibly bring the surfaces into contact at a finite speed, then this equilibration may not occur, as the water and ions may not diffuse out of the pore fast enough for the forces to converge to their equilibrium values. Free water molecules diffuse out of the pore more readily than the hydrated ions, and hence the concentration of ions in the pore will be enriched relative to the equilibrium concentration for that separation. Thus, the relative velocity of the particles generally produces increased repulsion with the relative speed of approach, as shown in Figure 17. 5. Discussion 5.1 Surface Structure Effects. Our experiments clearly demonstrate that the agglomerative bond is different for different faces of potash alum. This is quite a marked effect (Figure 8), and although the simulated surface structures (Figure 13) do show variations in atomic structure, all are composed of sulfates and hydrated cations. Thus, the existence and strength of any agglomerative bond can be expected to vary considerably with the atomic structure of the surface. The nature of the surface will considerably affect the last few nanometers of the collisions of the particles prior to the formation of any agglomerative bond. The variations in the ion distributions and double layer effects on the different faces of potash alum certainly imply considerable differences in the forces required for the faces to come into contact, let alone the likely growth rate of the agglomerative bond and its subsequent strength. Determining the exact structure of the surface
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Figure 17. The total forces on the nanocrystallites of KCl as a function of separation, as they move toward each other with various velocities. The crystallites can move laterally in the xy plane. The crystallites are in the simulation cell shown in Figure 14.
Figure 16. Snapshots of simulated configurations of solution molecules between KCl surfaces in close contact at a separation of 6.0 Å. (a) The water structure when the faces are held aligned, as in the crystal, enhances the attractive force between the surfaces. (b) When the crystallites are in a saturated solution of KCl, the majority of ions diffuse quickly out of the rigid pore. The one trapped potassium ion disrupts the water structure, reducing the attractive force resulting from the solvent ordering. (c) When the crystallite surfaces are mobile, then the surface and solvent ions can form an extremely stable structure, with little net force between the faces.
on the scale of the interacting faces will be difficult. Succinic acid (020) looks like an ideal case, where cutting the crystal structure results in a very stable looking surface (Figure 11). However, the observations reported here (Figure 4) indicate that it is not flat on the optical scale under the conditions of experimental growth, implying that the perfection of the modeled surface does not extend to this scale. 5.2 Nature of the Agglomerative Bond. The difference in the agglomerative bond strengths between different faces of potash alum almost certainly implies that the microscopic nature of the agglomerative bond itself differs between the faces. The agglomerative bond is unlikely to be completely amorphous and isotropic but will have partial crystallinity dictated by the surfaces involved. The agglomerative bridge could only be completely crystalline if we have like crystal surfaces in perfect register. In this case, or for other symmetryrelated alignments, practically distinguishing agglomeration via aggregation from twinned crystal growth is not yet possible. The urea example (section 4.2) does
show that unlike faces can form quite attractive contacts, albeit with the zone axis aligned. This suggests that the range of collision orientations of two surfaces that may be suitable for the formation of an agglomerative bond with some degree of crystallinity could be quite large. A systematic analysis of the relative orientations of crystals within agglomerates would provide considerable insight into the mechanism of agglomeration, which would be deepened by nanoscale characterization of the nature of the agglomerate bond. The dependence of the intercrystal forces on the relative orientation of even parallel surfaces could be very large.60 Contrasting the model surfaces for KCl, potash alum, and succinic acid also shows that the surface structure introduces geometric factors that will be very dependent on the type and face of the crystal. The approaching crystals will experience lateral forces that will tend to optimize their alignment. For highly symmetric faces with small component ions, such as KCl and potash alum, such alignment requires displacements of only a few atoms width, which may readily occur during collision. This is less likely for surfaces containing larger molecules, which will require more movement to improve the alignment. One factor is clear from the potash alum experimental results (Figure 7b): this agglomerative bond is not a single neck between the particles that grows outward. It must be recalled that in our experiments, the agglomerative bonds grew under conditions where the faces were held parallel and left for many minutes for the bonds to form. These are hardly the conditions that pertain in crystallizers! However, the obvious inclusion of solvent within both the potash alum and succinic acid agglomerative bonds (Figure 7b,d) does suggest that the behavior of solvent confined to narrow slit pores must play an important role in forming these bonds. The simulations for KCl suggest that the attempted departure of some ions from between the faces to maintain equilibrium may assist the growth of agglomerative bonds. In the case of the perfectly flat surfaces of potash alum, this appears to have led to the formation of a rim. Once the rim has formed this will block diffusion of ions into the contact area, thereby trapping solvent in the smooth cavities. The rougher faces of succinic acid not
Perspective
surprisingly lead to a very irregular distribution of agglomerate bonds and trapped solvent. Since the agglomerative bonds only form in part of the overall contact area, the forces measured for their disruption are very much smaller than those that would be required to form a complete 2D surface for the contact area. The theoretical maximum agglomerative bond strengths are estimated by surface energies or attachment energies. 5.3 Hydrodynamic Effects. The speed of the crystal collision may have a significant effect on the probability of the collision leading to agglomeration, since the forces involved appear very dependent on the speed of approach. The speed of approach relative to the time for the ions to diffuse out of the slit pore, together with the area of the slit pore, will affect whether the solution is in equilibrium or not. This sheds some doubt on whether simulations at constant chemical potential are appropriate for quantifying crystal agglomeration. The power dissipation within large crystallizers is highly spatially distributed, leading to local variations in collision velocities. These collision velocities can easily exceed those simulated in Figure 17. Hence, there is a strong implication that the local velocity will have a major effect on the forces at close approach and the nonequilibrium distribution of ions between the approaching crystallites. Therefore, the local velocity will be a significant factor in the aggregation efficiency. 5.4 Effect of Supersaturation. The experimental results clearly show that the degree of supersaturation has a major effect on the final strength of the bond. This is probably due, predominantly, to the supersaturation affecting the rate of growth of the agglomerative bond. However, the simulations do show that the ion concentration will also affect the nature of the ion distribution in the surface/solvent interface area (in a different way for different faces). This will affect the forces during aggregation, through the rearrangements necessary to accommodate the solvent molecules as the faces approach through the last nanometers of the collision. 5.5 Forces Involved in Aggregation. The simulations imply that the forces between two crystallites approaching each other will be very dependent on the geometry of the collision and the degree to which the speed of the collision and structure of the surface allow the relative orientations to be optimized and the solvated ions to equilibrate. The structure of the solution between the faces can give rise to strong attractive forces when there is compatibility between the surfacesurface, solvent-solvent, and solvent-surface interactions, such as in the cases illustrated by Figure 16a,c. Although the forces can vary considerably with separation and may often be repulsive, crystals may well be able to stay in some of these very stable arrangements for long enough for an agglomerative bond to start to form (i.e., such structures explain the stickiness of some collisions). The simulations for small pore slits show many configurations of crystallites and solvent ions that could be persistent “preagglomerative bonding” arrangements. Agglomeration definitely does not require microscopically flat surfaces in perfect alignment. Indeed, a surface that is rough on the optical scale, such as succinic acid, could well have many microscopically flat areas that
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attract on contact, like the hairs on a gecko’s foot. This would be consistent with analysis of agglomerates showing a range of crystal alignments. 6. Conclusions Our studies raise many important questions that require investigation before we can hope to understand agglomeration sufficiently to predict an aggregation rate or an agglomerate disruption rate for a specific crystallizing system. The computer modeling of the surfaces involved and the forces between approaching crystallites, described here, show that many factors need to be considered to quantify the dynamics of collisions of crystals in crystallizers over the last few nanometers that are the key to aggregation. Although we can see many possibilities for these collisions to allow the particles to stick together for a while, it is beyond current capabilities to simulate the growth of the agglomerative bond. Quantifying this would be central to predicting the efficiency with which collisions will result in an agglomerative bond, i.e., quantifying the aggregation kernels (section 2.1). This will require both experimental and simulation studies to define the molecular level mechanisms of the bond formation and growth. The disruption of the agglomerative bond is experimentally accessible and the experimental measurements of force required to break that bond have been used to predict the fragmentation particle size, thus providing a major input into the disruption kernel. However, the prediction of the strength of an agglomerative bond requires far more understanding of its microscopic nature and the relationship to the surfaces involved. Although considerable work is required to quantitatively understand agglomeration, there has been significant progress in both experimental and computer simulation studies of idealized surfaces and confined systems over the past few decades. Hence, despite the complexity of industrial crystallization processes, sufficient interdisciplinary cooperation could well lead to a molecular scale understanding and predictive capability. Acknowledgment. This project was funded by the EPSRC under the “Collaboration between Chemists and Chemical Engineers” programme Grant GR/M73156/01. References (1) Jones, A. G. Anal. Proc. 1993, 30, 456-457. (2) Sohnel, O.; Mullin, J. W.; Jones, A. G. Ind. Eng. Chem. Res. 1988, 27, 1721-1728. (3) Wachi, S.; Jones, A. G. Chem. Eng. Sci. 1992, 47, 31453148. (4) Beckman, J. R.; Farmer, R. W. AIChE Symp. Ser. 1978, 8, 85-94. (5) Hostomsky, J.; Jones, A. G. J. Phys. D Appl. Phys. 1991, 24, 165-170. (6) Tai, C. Y.; Chen, P. C. AIChE J. 1995, 41, 68-77. (7) Hartel, R. W.; Gottung, B. E.; Randolph, A. D.; Drach, G. W. AIChE J. 1986, 32, 1176-1185. (8) Hartel, R. W.; Randolph, A. D. AIChE J. 1986, 32, 11861195. (9) Wojcik, J. A.; Jones, A. G. Chem. Eng. Res. Des. 1997, 75, 113-118. (10) Wojcik, J. A.; Jones, A. G. Chem. Eng. Sci. 1998, 53, 10971101.
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(11) Pratola, F.; Simons, S. J. R.; Jones, A. G. Chem. Eng. Res. Des. 2002, 80, 441-448. (12) Klein, J.; Kumacheva, E. J. Chem. Phys. 1998, 108, 69967009. (13) Smoluchowski, M. V. Z. Phys. Chem. 1917, 19, 129-168. (14) Low, G. C. Agglomeration Effects in Aluminium Trihydroxide Precipitation; Low, G. C., Ed.; University of Queensland: Australia, 1975. (15) Mumtaz, H. S.; Hounslow, M. J.; Seaton, N. A.; Paterson, W. R. Chem. Eng. Res. Des. 1997, 75, 152-159. (16) Mersmann, A.; Braun, B. Agglomeration; 2nd ed.; Mersmann, A.; Braun, B., Eds.; Dekker: New York, 2001. (17) Hounslow, M. J.; Mumtaz, H. S.; Collier, A. P.; Barrick, J. P.; Bramley, A. S. Chem. Eng. Sci. 2001, 56, 2543-2552. (18) Mersmann, A.; Braun, B.; Loffelmann, M. Chem. Eng. Sci. 2002, 57, 4267-4275. (19) Schubert, H. Int. Chem. Eng. 1981, 21, 363-377. (20) Pepin, X.; Simons, S. J. R.; Blanchon, S.; Rossetti, D.; Couarraze, G. Powder Technol. 2001, 117, 123-138. (21) Synowiec, P.; Jones, A. G.; Shamlou, P. A. Chem. Eng. Sci. 1993, 48, 3485-3495. (22) Thompson, P. D. A Transformation of the Stochastic Equation for Droplet Coalescence; Thompson, P. D., Ed.: Toronto, Canada, 1968; pp 1115-1126. (23) Adler, P. M. J. Colloid Interface Sci. 1981, 83, 106-115. (24) Camp, T. R.; Stein, P. C. J. Boston Soc. Civ. Eng. 1943, 30, 219. (25) Jones, A. G Crystallization Process Systems; ButterworthHeinemann: Oxford, 2002. (26) Mullin, J. W. Crystallization; 3rd ed.; Butterworth-Heinmann: Oxford, 1993. (27) Zauner, R.; Jones, A. G. Chem. Eng. Sci. 2000, 55, 42194232. (28) Simons, S. J. R.; Pratola, F.; Jones, A. G.; Brunsteiner, M.; Price, S. L. Part. Part. Syst. Charact., in press. (29) Simons, S. J. R.; Fairbrother, R. J. Powder Technol. 2000, 110, 44-58. (30) Mullin, J. W.; Zacek, S. J. Cryst. Growth 1981, 53, 515518. (31) Jones, A. G.; Mydlarz, J. Can. J. Chem. Eng. 1990, 68, 250259. (32) Nyvlt, J.; Karel, M.; Pisarik, S. Cryst. Res. Technol. 1994, 29, 409-415. (33) Pamplin, B. R. Crystal Growth; Pergamon Press: Elmsford, NY, 1980. (34) Tolansky, S. An Introduction to Interferometry; Longman Group Ltd:, 1955. (35) Jancic, S. J. Crystallization Kinetics and Crystal Size Distribution in Mixed Suspension Mixed Product Removal Crystallizers; Jancic, S. J., Ed.; University College London, 1976. (36) Ilievski, D.; White, E. T. Chem. Eng. Sci. 1994, 49, 32273239.
Perspective (37) David, R.; Marchal, P.; Klein, J. P.; Villermaux, J. Chem. Eng. Sci. 1991, 46, 205-213. (38) Bramley, A. S.; Hounslow, M. J.; Newman, R.; Paterson, W. R.; Pogessi, C. Chem. Eng. Res. Des. 1997, 75, 119-124. (39) Addai-Mensah, J.; Prestidge, C. A.; Ralston, J. Miner. Eng. 1999, 12, 655-669. (40) Fasoli, U.; Conti, R. Kristall. Technol. 1973, 8, 931-946. (41) Crawley, G. M.; Gruy, F.; Cournil, M. Chem. Eng. Sci. 1996, 51, 4537-4550. (42) Donnay, J. D. H.; Harker, D. Am. Minerol. 1937, 22, 446. (43) Hartman, P.; Bennema, P. J. Cryst. Growth 1954, 8, 4952. (44) Berkovitch-Yellin, Z. J. Am. Chem. Soc. 1985, 107, 82398253. (45) Brunsteiner, M.; Price, S. L. Cryst. Growth Des. 2001, 1, 447-453. (46) Hartman, P.; Bennema, P. J. Cryst. Growth 1980, 49, 145146. (47) Amara, N.; Ratsimba, B.; Wilhelm, A. M.; Delmas, H. Ultrason. Sonochem. 2001, 8, 265-270. (48) Engkvist, O.; Price, S. L.; Stone, A. J. Phys. Chem. Chem. Phys. 2000, 2, 3017-3027. (49) Reyhani, M. M.; Freij, S.; Parkinson, G. M. J. Cryst. Growth 1999, 198-199, 258-263. (50) Brunsteiner, M.; Price, S. L. J. Phys. Chem. B 2004, 108, 12537-12546. (51) Gale, J. D.; Rohl, A. L. Mol. Simulat. 2003, 29, 291-341. (52) Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P. J. Phys. Chem. 1987, 91, 6269-6271. (53) Allan, N. L.; Rohl, A. L.; Gay, D. H.; Catlow, C. R. A.; Davey, R. J.; Mackrodt, W. C. Faraday Discuss. 1993, 273-280. (54) Sakuntala, T.; Arora, A. K.; Shekar, N. V. C.; Sahu, P. C. J. Phys.-Condens. Mater. 2000, 12, 4417-4432. (55) Smith, W.; Forester, T. R. J. Mol. Graphics 1996, 14, 136141. (56) Rudolph, W. W.; Mason, R.; Pye, C. C. Phys. Chem. Chem. Phys. 2000, 2, 5030-5040. (57) Chang, T. M.; Dang, L. X. J. Phys. Chem. B 1999, 103, 4714-4720. (58) Hansen, J. P.; Lowen, H. Annu. Rev. Phys. Chem. 2000, 51, 209-242. (59) Stone, A.; Dullweber, A.; Hodges, M. P.; Popelier, P. L. A.; Wales, D. J. ORIENT; 3.2 ed.; Stone, A., Dullweber, A., Hodges, M. P., Popelier, P. L. A., Wales, D. J., Ed.; University of Cambridge, 1996. (60) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1991. (61) Yang, K. L.; Yiacoumi, S.; Tsouris, C. J. Chem. Phys. 2002, 117, 8499-8507. (62) Brunsteiner, M.; Price, S. L. Langmuir, in preparation.
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